Finite Group Databasebeta

← #28362959

Subgroups — #28362959

Order 13952.

584 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 101 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×#13182030C211yesmaximal, characteristic
2C2×#16812618C211yesmaximal, characteristic, complemented
3C2×#16812615C212yesmaximal, complemented
4#20294989C212yesmaximal, complemented
5C2×#101×C109C211yesmaximal, characteristic, complemented
6C2×#13182030—24no—
7C2×#13182030—24no—
8C2×#13182030—22no—
9C2×#13182030—22no—
10C22×D436C412yes—
11C22×D436C2211yescharacteristic
12C22×C4×D109C412yes—
13C22×C4×D109C2211yescharacteristic
14C2×#13182013C2212yes—
15#16812618—22no—
16#16812615—24no—
17#16812611—24no—
18C22×Q8×C109C2211yescharacteristic
19C2×M16×C109C2212yes—
20#101×C109—22no—
21C23×D109C2×C412yes—
22C23×D109C2×C411yescharacteristic
23#13182030—216no—
24#13182030—216no—
25C2×D436—24no—
26C2×D436—24no—
27C2×D436—24no—
28C2×D436—22no—
29C2×D436—21no—
30C2×D436—21no—
31C2×D436D414yes—
32C22×Dic109C2×C411yescharacteristic
33C2×C4×D109—24no—
34C2×C4×D109—24no—
35C2×C4×D109—22no—
36C2×C4×D109—21no—
37C2×C4×D109—21no—
38C2×C4×D109—24no—
39C2×C4×D109D414yes—
40#13182013—22no—
41#13182013—22no—
42C2×#11724784—22no—
43C2×Q8×C109—22no—
44C2×Q8×C109—22no—
45C2×Q8×C109D412yes—
46C2×Q8×C109D412yes—
47C22×C4×C109C2×C412yes—
48C22×C4×C109C2311yescharacteristic
49M16×C109—22no—
50M16×C109—22no—
51C2×C8×C109—22no—
52C23×C109C22×C411yescharacteristic
53C22×D109—21no—
54C22×D109—22no—
55C22×D109—22no—
56C22×D109—21no—
57C22×D109—42no—
58C22×D109—44no—
59C22×D109C22⋊C412yes—
60C22×D109C22⋊C414yes—
61Q8×C109—24no—
62Q8×C109—24no—
63C2×C4×C109—22no—
64C2×C4×C109—22no—
65C2×C4×C109C22⋊C414yes—
66C2×C4×C109C2×D412yes—
67C2×C4×C109C2×D412yes—
68C2×C4×C109C2×D411yescharacteristic
69C2×C4×C109C2×D411yescharacteristic
70C2×Dic109—21no—
71C2×Dic109—21no—
72C2×Dic109C22⋊C412yes—
73D436—216no—
74D436—24no—
75D436—24no—
76C4×D109—216no—
77C4×D109—24no—
78C4×D109—24no—
79C8×C109—42no—
80#11724784—42no—
81C22×C109—22no—
82C22×C109C2×C22⋊C411yescharacteristic
83C22×C109C2×C22⋊C411yescharacteristic
84C22×C109C2×C22⋊C411yescharacteristic
85C2×D109—22no—
86C2×D109—24no—
87C2×D109—24no—
88C2×D109—42no—
89C2×D109—42no—
90C2×D109—44no—
91C2×D109—44no—
92C2×D109—44no—
93C4×C109—24no—
94C4×C109#12012yes—
95C4×C109#12012yes—
96Dic109—22no—
97C2×C109—21no—
98C2×C109—21no—
99C2×C109#37911yescharacteristic
100C2×C109#41312yes—
101D109—42no—
102D109—44no—
103#1785—1091nomaximal, Sylow
104C109#178511yescharacteristic, complemented, Sylow
105C22×#41—1091no—
106C2×#101—1091no—
107C2×#101D10911yescharacteristic, complemented
108C2×#100—1092no—
109#407—1092no—
110C2×#41—2184no—
111C2×#41—2184no—
112C2×#41—2182no—
113C2×#41—2182no—
114C22×Q8C2×D10911yescharacteristic
115C22×D4—1092no—
116C22×D4—1091no—
117C23×C4—1091no—
118C23×C4—1092no—
119C2×M16—1092no—
120C2×M16C2×D10912yes—
121#101—22no—
122#101—2182no—
123#100—2184no—
124C22⋊C8—2184no—
125C24—1092no—
126C24—1091no—
127#41—21816no—
128#41—21816no—
129C2×Q8—22no—
130C2×Q8—22no—
131C2×Q8D43612yes—
132C2×Q8C22⋊D10912yes—
133C2×D4—1094no—
134C2×D4—2184no—
135C2×D4—2184no—
136C2×D4—2184no—
137C2×D4—2182no—
138C2×D4—2181no—
139C2×D4—2181no—
140C22×C4—1091no—
141C22×C4—1094no—
142C22×C4—2184no—
143C22×C4—2184no—
144C22×C4—2184no—
145C22×C4—2182no—
146C22×C4—2181no—
147C22×C4—2181no—
148C22×C4C4×D10912yes—
149C22×C4C22×D10911yescharacteristic
150M16—22no—
151M16—22no—
152M16—2182no—
153M16—2182no—
154C2×C8—22no—
155C2×C8—2182no—
156C23—1094no—
157C23—1092no—
158C23—4364no—
159C23—4362no—
160C23—2181no—
161C23—2182no—
162C23—2182no—
163C23—2181no—
164C23C2×C4×D10911yescharacteristic
165Q8—24no—
166Q8—24no—
167D4—21816no—
168D4—2184no—
169D4—2184no—
170C2×C4—22no—
171C2×C4—22no—
172C2×C4—1092no—
173C2×C4—21816no—
174C2×C4—2181no—
175C2×C4—2181no—
176C2×C4—2184no—
177C2×C4—2184no—
178C2×C4#1318200614yes—
179C2×C4D4×D10912yes—
180C2×C4D4×D10912yes—
181C2×C4C2×C22⋊D10911yescharacteristic
182C2×C4C2×D43611yescharacteristic
183C8—42no—
184C8—4362no—
185C22—22no—
186C22—4364no—
187C22—4362no—
188C22—4364no—
189C22—4362no—
190C22—4364no—
191C22—2184no—
192C22—2184no—
193C22—2182no—
194C22C22⋊C4×D10911yescharacteristic
195C22C22⋊C4×D10911yescharacteristic
196C22C2×#1318200611yescharacteristic
197C4—24no—
198C4—2182no—
199C4#1681266712yes—
200C4#1681266912yes—
201C2—21no—
202C2—21no—
203C2—4364no—
204C2—4362no—
205C2#2029488111yescharacteristic
206C2#2029500912yes—
subgroupG/H
centerC22C22⋊C4×D109
commutatorC23×C109C22×C4
FrattiniC22×C4C22×D109
FittingC2×#101×C109C2
socleC22×C109C2×C22⋊C4
prime radicalC22×#13182030C2
cosocleC22×C4×C109C23
hypercenterC2×#101D109
hypocenterC109#1785
LCSG ▷ C23×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#101

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#16812618C2semidirect
3. C2×#16812615C2semidirect
4. #20294989C2semidirect
5. C2×#101×C109C2semidirect
104. C109#1785semidirect
107. C2×#101D109semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C22×#131820302
C2×#168126182
C2×#168126152
#202949892
C2×#101×C1092
#1785109